In this video I take a look at how to find the Volume between two surfaces using double integration and polar coordinates. I first look at defining the region of intersection between the two surfaces. I then demonstrate how to convert the region from cartesian (x, y) coordinates to polar coordinates (r, Θ).
I then look in detail at the process of integrating in the r direction (the inner integral) using diagrams to visual the process, and also explain how the limits of integration are obtained. Moreover, I look at how infinitesimally small pieces of area, dA, are calculated using rdrdΘ. I then extend this to show how infinitesimally small pieces of volume are obtained by multiplying the area by a height component f(x, y) to give f(x, y)rdrdΘ.
I then move on to the outer integral and ask the viewer to visualize the process with the help of animated diagram. I show how the the radial distance rotates about the origin from Θ =0 to Θ = 2π.
Having define the integrals I demonstrate how to determine the height component how to convert it into polar coordinates.
Finally, I evaluate the integrals.